# What is Rigid body dynamics: Definition and 68 Discussions

In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. The assumption that the bodies are rigid (i.e. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. This excludes bodies that display fluid, highly elastic, and plastic behavior.
The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law (kinetics) or their derivative form, Lagrangian mechanics. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a function of time. The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems.

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1. ### Friction Required for Billiard Ball to Roll without Slipping

[Mentor Note: Two threads merged below and the OP has fixed up their LaTeX in later posts] TL;DR Summary: A billiard ball with zero angular velocity and linear velocity v_0 is released on a horizontal surface with coefficient of friction mu_k. It begins to roll without slipping after...
2. ### A Rigid body motion (RBM) transformation

Above are pictures of a problem in mechanics. An elastic body occupying domain ##\Omega## is supported below with a fixed support and from above with a rigid body. The following calculations aim to express the movements of the points located at the interface ##\Gamma_F## between the rigid and...
3. ### I Frame Transformation in rigid bodies

I'm using rigid body dynamics/kinematics in robotics stuff but I don't have a background in mechanics, I'm interested in understanding the kinematics of frame transformations for rigid bodies. Suppose we have two reference frames fixed on a rigid body, F_1 and F_2 and a transformation T which...
4. ### Angular momentum of rotating hoop

Problem: Official solution: My calculation: \begin{align*} \mathbf L &= M \mathbf R \times \mathbf V + \mathbf L_{cm} \\ &= M R (\hat j + \hat k) \times (- \Omega R \hat i) + MR^2 \Omega \hat j \\ &= MR^2 \Omega (\hat k - \hat j + \hat j) \\ &= MR^2 \Omega \hat k \end{align*}
5. ### Understanding Kinetic Energy: Moment of Inertia and Rotational Motion

Consider the above setup. Here, to get the kinetic energy of the body, the moment of inertia with respect to the ##y-##axis has to be calculated. This can be done in two ways: 1. The moment of inertia of the rotation around the center of mass is ##\Theta_s##, then the kinetic energy is...
6. ### I Intuition Behind Intermediate Axis Theorem in an Ideal Setting

For a rigid body with three principal axis with distinct moments of inertia, would the principal axis with the intermediate moment of inertia still be unstable in ideal conditions, e.g. no gravity, no friction etc.? From the mathematical derivation I deduce that it should be unstable, since we...
7. ### Billiards with rectangular prisms

I plan to add on to this as I have time and ability. Apologies for the weird formatting.
8. ### B Does a body behave as a point mass even at rest?

Hi, A body with center of mass behaves as a point mass when a force is applied. So when ##F_{ext}=0## then does it also behave as a point mass with ##a_{com}=0##, at rest. If yes, How can we prove this? (And can somebody please answer my other question I posted a week ago...
9. ### A spring, disk and pulley system

(a) By setting up a coordinate system with the x-axis pointing to the right and the y-axis pointing downward we have ##\begin{cases}-kx_{eq}+T_1+F_{s}=0\\ -RF_{s}+rT_1=0\\ r_p (T_2-T_1)=0\\ -T_2+mg=0\end{cases}\Rightarrow x_{eq}=\frac{mg}{k}\left(1+\frac{r}{R}\right)## which coincides with the...
10. ### I A Question on Spinors in a High school textbook

While revising Rotational motion, I came across a qualitative question which blew me away. Meaning I couldn't even understand the question let alone answer it😅. It has to do with these objects called spinors which as I understand are evoked in quantum mechanics and Relativity. I am attaching the...
11. ### A doubt in the rotational analogue of F=ma

If we have a cylinder rolling on the ground ##\tau = I\alpha## can only be applied about the point in contact with the surface(Instantaneous axis of rotation) and its CoM I don't see why this should be the case. why can the equation ##\tau = I\alpha## only be applied about the axis passing...
12. ### Vertical beam on a frictionless surface

This could also be posted in the Math / differential equations sub, but it also involves the derivation which is classical physics. So I was doubting :smile:. So, I'm dusting off my dynamics a bit and found this problem of a thin beam on a frictionless surface in a different forum and decided...
13. ### How to calculate hinge forces?

I have solved problem (A). But I couldn't solve problem (B). Here is my attempt:
14. ### Determining the Moment of Inertia about an angle θ to the x axis

I thought about solving it using components of IAB but since it is a scalar quantity it doesn't seems to be correct . I don't think Perpendicular Axis theorem will work as required Inertia is in the same plane.
15. ### Euler Equations for Dynamics of rigid body

I have been studying the dynamics of free top from Morin's book. In his book when describing the dynamics,he writes down the equation of motion as shown in screenshot. However,I am not able to understand which term refers to which coordinate system. For eg: Here ##\omega## refers to angular...
16. ### Questions Regarding the Inertia Tensor

In Chapter 11: Dynamics of Rigid Bodies, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, pages 415-418, Section 11.3 - Inertia Tensor, I have three questions regarding the Inertia Tensor: 1.The authors made the following statement: "neither V nor ω...
17. ### Cylinder with Displaced Center of Mass Rolling Down Incline

Homework Statement A rigid cylinder of radius ##R## and mass ##\mu## has a moment of inertia ##I## around an axis going through the center of mass and parallel to the central axis of the cylinder. The cylinder is homogeneous along its central axis, but not in the radial and angular directions...
18. ### Find the angle for which both disks are no longer in contact

Homework Statement A mobile disk of radius R and mass M is moving above another immobile vertical disk of the same radius. Initially the mobile disk is at rest at the highest point above the immobile disk and then it starts rolling without slipping. Assuming the mobile disk never slips, find...
19. ### Pure rolling of sphere having non uniform mass density ?

in case of rolling without slipping of a solid sphere having uniform mass density the condition is Vcm (velocity of center of mass ) = Rω or [a][/cm] = Rα ,which comes from the fact that if an object that rolls without slipping the geometric center of the body travels 1 circumference along...
20. ### IC Engine Connecting Rod Mass Moment of Inertia Measurements

Does anyone know of a source, on the Internet preferably, for component inertial data for a real IC engine? In particular, what I'd like to obtain (for some real engine, any engine) are these items: 1. Piston weight (including wrist pin) 2. Cylinder bore dimension 3. Connecting rod length...
21. ### Rigid Body Dynamics in CAD/CAE

Hi folks, This week I have a question about rigid body dynamics analyses in CAE. This is in relation to the final year project of my degree. Over the past few months I've been working with AnSys 18.1 for FEA however, I've reserved AnSys exclusively for static analyses. For rigid body dynamics...
22. ### Motor driven hydraulic cylinder design

I am trying to design a mechanism which has a hydraulic cylinder driven by a step motor. The motor shaft is connected to the hydraulic cylinder piston via two rods of lengths r and L as shown in the below figure. I want the cylinder's piston to be driven at constant velocity v, so I am trying to...
23. ### How Does the Angle of Inclination Affect Motion in Rigid Body Dynamics?

Homework Statement In the figure there are a block 1 of mass ##M##, a pulley (disk) of mass ##M## and radius ##R## and a second block of mass ##2M## on an inclined plane with an angle ##\alpha## with respect to the horizontal. On this plane the static friction is ##\mu##. The wire does NOT slip...
24. R

### Rigid body relative acceleration

Homework Statement Rigid body (ship) rotate and moves around 3 axis (x,y,z) around the center of gravity. The position is of the center of gravity is not known. What is known: At a point (A) the accereleration, velocity and position and rotational acceleration, velocity and position are known...
25. P

### I Instability of a Rigid Untethered Ring Around a Planet

I had this discussion while driving home to California from a trip to Washington state with a friend. We were discussing the stability of a completely rigid, untethered, ring-structure around Earth, and I did not know how to explain to him that such a thing must be tethered by rigid towers lest...
26. ### Equations of Motion: Constrained Hybrid Dynamics

Hi all, I've tried to figure this out for some time without luck. Hope you might be able to give some input. I've implemented a model-based dynamics software in MATLAB based on the works of Roy Featherstone's Springer book "Rigid Body Dynamics Algorithms". So, I have the EoM of an...
27. ### 2 questions from rolling motion

Both the questions are at https://imgur.com/a/o7B20 The problem I am facing is that on a straight line friction is there and I can't balance it. Using F=ma and T=Iw doesn't work.
28. ### Finding the force of constraint--compound pendulum on spring

Homework Statement From Fetter and Walecka 5.1:[/B] Consider the compound pendulum in FIg 28.1 (mass M, moments of inertia Iij relative to the center of mass, which is a distance L from the point of support Q) but with Q attached to the bottom of a vertical spring (force constant k) and...
29. ### Is the Proof Behind Choosing Any Point on an Axis to Calculate Torque?

I know that to calculate the torque about an axis, we can choose any point on that axis and find the torque about that point and take the component along the direction of the axis.Buy what is the proof behind this theorem?
30. ### Stuck with 2D kinetics problem

Homework Statement Hi, I've been trying to solve this exercise for the last four hours and I'm totally stuck. The problems goes like this: Given the mechanism in the image, located in the vertical plane, the OB cord is cut when t=0. Find the reactions Rx, Ry and N in {Kgf} for the instants...
31. ### A doubt from rigid body dynamics

in rigid body dynamics is this relation between velocity of centre of mas and velocity of any particle in the rigid body correct? vcm = vp + rXω r is the position vector of the particle with respect to centre of mass relations are written in vector form and also tell me how to calculate...
32. ### A problem in rigid body dynamics

Homework Statement A disc of mass M and radius r is kept on a horizontal,frictional plane and is connected to a horizontal spring at the centre.A particle of mass m strikes the topmost point of the disc,tangentially and sticks to it.Assume that the mass of the particle is m and it's velocity is...
33. ### Moment of thin Rod at pivot when acted upon by a Force

This is of my own interest/ practice. Homework Statement A thin rod (of width zero, but not uniform) is pivoted freely at one end about the horizontal z axis , being free to swing in the xy plane (x horizontal, y vertically down). Its mass is m and its CM is a distance a from the pivot. The...
34. ### Can I Find the Moment of Inertia with Only 8 Equal Parts in My Homework Problem?

Homework Statement http://i.imgur.com/7ZRsVj5.png Homework Equations I can't apply correct method ! The Attempt at a Solution In bigger figure, i cut into 8 equal parts.So , total moment of inertia=8 I But it's wrong ! I know it's wrong because i can't able to find angle between any two...
35. ### Predict the position of a particle on a rigid body

1. Problem Statement Assume there is an rigid object with mass m in 2D space, an impulse J = FΔt is applied at time t1 at the particle Pimp and Pimp is on the exterior boundary of the object. The impulse cause a free plane motion of the object and the object is only affected by the force of the...
36. ### Rotation of a rigid body about external axis

in the figure a rigid body - a circle- is moving such that its centre is moving in a circular path but the orientation of the body is fixed with respect to the centre of the body (the circle). According to def of rotion of rigid body - Rotation of a rigid body about a fixed axis is defined as...
37. ### Time taken to start pure rolling

Homework Statement A solid sphere of radius R is set into motion on a rough horizontal surface with a linear speed v0 in forward direction and angular speed ω0##=\frac{v_0}{2R}## in counter clockwise direction. Find time after which pure rolling starts. Homework Equations For pure rolling...
38. ### Lagrange equation of motion for tensegrity

Hi, I have read this paper “Dynamic equations of motion for a 3-bar tensegrity based mobile robot” (1) and this one “Dynamic Simulation of Six-strut Tensegrity Robot Rolling”. 1) http://digital.csic.es/bitstream/10261/30336/1/Dynamic%20equations.pdf I am trying to implement a tensegrity...
39. ### Total derivative involving rigid body motion of a surface

This stems from considering rigid body transformations, but is a general question about total derivatives. Something is probably missing in my understanding here. I had posted this to math.stackexchange, but did not receive any answers and someone suggested this forum might be more suitable. A...
40. ### Finding acceleration of block connected to pulley with mass

Homework Statement Homework Equations a=Rα The Attempt at a Solution Free body diagrams: For larger pulley, TR=Iα ⇒ T=½MRα If this pulley rotates by some amount suchat x length of rope becomes free, and if I hold the smaller pulley at rest, then this extra length of rope comes bellow the...
41. ### Rigid body dynamics: falling stick

Homework Statement A stick of length L and mass m is falling on a table with kinetic friction uK and static friction uS. Find the equation that determines whether the rod will stick or slip. If m = 10kg and L = 100cm, uS = 0.8, uK = 0.7 and initial theta = 30 degrees find the initial accel for...
42. ### A homework assignment including rotation of a rigid body

Homework Statement The problem is this: A car is evenly slowing down from 30 km/h to 0 in the time of 2 seconds. Radius of its wheels is 30 cm. What is its angular acceleration and what total angle will the wheel describe until it stops? How many turns does the wheel make and what is the length...
43. ### Direction of instantaneous axis in rigid body dynamics

The general motion of a rigid body can be considered to be a combination of (i) a motion of its centre of mass about an axis, and (ii) its motion about an instantaneous axis passing through the centre of mass. These axes need not be stationary. Consider, for example, a thin uniform disc welded...
44. ### Conserving Angular Momentum (Rigid Body Dynamics)

Lets say i have a rod (length = L) hinged at one end (A).It is initially at rest.Now if an impulse (J) acts on the other end (B),can i conserve the angular momentum about A(the hinge)? that is can i write: JL=Iw?(I=moment of inertia,w=angular velocity) this is what i saw in the book. My Doubt...
45. ### 2D Rigid Body Dynamics - Newtonian Equaitons of Motion

Homework Statement In the diagram attached a bar weighs 500 kg and is a 2D plane rigid body with mass centre at G. At the instant shown, the bar is moving horizontally at but reducing speed, causing horizontal deceleration as shown by vector a=-7i m/s2. At the same time, the bar is being...
46. ### How can I calculate the depth of bullet penetration in different materials?

Hey guys; I am new to the forums. I am an effects artist who's working a Rigid Body Dynamics project and I may be able to use some of you guys' expertise while I'm building these simulations. As I stated above, I'm doing R&D for a deformation system I'm working on. What I want to do is...
47. ### Understanding 3D Rigid Body Dynamics Equations: R=R'Ω Explained

Homework Statement I just saw the equation R=R'Ω (where ' is dot, or the derivative) in my notes (unorganized obviously), and can't remember what it stands for (rotation matrices?). Can somebody please tell me what this equations means thanks. I know R is some position vector and Ω is the...
48. ### Stuck on derivation of Euler's equations in rigid body dynamics

i was reading about derivation of Euler's equations for rotational dynamics (john taylor, classical mechanics, chapter 10) when i got stuck on one of the reasonings essentially it refers to the moment of inertia tensor, since the tensor itself about a point is dependent on the position of the...
49. ### 3D Rigid body dynamics - Rod over rotating disk

Homework Statement A thin uniform rod is attached to an axis through its midpoint. The axis is standing on a disk rotating with constant angular speed \Omega about its symmetry axis. The rod's midpoint is located directly above the rotational axis of the disk. Let \theta denote the rod's...
50. ### Solve Rigid Body Collision: System Approach w/Cons. Angular Momentum

under what conditions can a rigid body collision problem be solved using a system approach, (i.e by using the conservation of angular momentum of the two rigid bodies about some point) the equation M=d[H]/dt is only valid when M and H are taken about a point fixed in a massless extension of a...